Lipschitzian SLLNs for random functions
2026-07-22 • Machine Learning
Machine Learning
AI summaryⓘ
The authors prove that averages of certain types of functions, called locally Lipschitz functions, behave predictably under specific conditions related to topology or logic. Their approach covers a wide range of functions, including those definable in special mathematical systems called o-minimal structures. They show that important properties like convergence of derivatives and identifying solutions from limited data hold reliably for these functions. This work addresses and overcomes some problems they found in their earlier research.
strong law of large numberslocally Lipschitz functionsLipschitz pseudometrico-minimal structurestopological conditionsmodel theoryClarke subdifferentialuniform convergencefinite-sample identificationdefinable functions
Authors
Lai Tian, Johannes O. Royset
Abstract
We prove strong laws of large numbers for locally Lipschitz functions in the Lipschitz pseudometric. Our results hold under either a topological or a model-theoretic condition, with the latter encompassing functions jointly definable in o-minimal structures but extending substantially beyond this class. Applications include uniform convergence of limiting and Clarke subdifferentials and finite-sample identification of solutions. Consequently, we identify broad classes of functions for which the failure phenomena revealed by our previous negative results [Tian and Royset, arXiv:2511.16568, 2025] do not occur.